There wasn't anything convoluted at all and no real legal proceedings.

Elizabeth wasn't born out of wedlock (though possibly conceived out of it) but the marriage was annulled following Anne Boleyn's execution in 1536. This put Elizabeth in the same position as her older half sister, Mary, with both being bastards according to the English church. This came after every single English subject had been forced to take the oath of succession, in which they swore to uphold the claim of any of Anne's issue over those of Katherine.

None of this mattered as Edward was born to Jane Seymour in 1537, which as a male heir trumped the act of succession anyway. Both Mary and Elizabeth remained the King's daughters but with uncertain status. Mary particularly spent some time dancing with death over her refusal to take the oath of supremacy- some historians argue that Cromwell persuaded her to do so and thus enabled her to reconcile with her father. The death of her mother also facilitated something of a reproachment, that source of antagonism between them having now gone.

Ultimately, Katherine Parr did a lot to bring both girls back into the royal fold in the 1540's, convincing Henry to have them for Christmas and encouraging Henry to enjoy his role as father and head of a family.

As Henry declined in the 1540's, he inevitably looked to the succession. This was easy- his son would succeed him. However, Henry being Henry, he sought to have unprecedented influence on the line of succession from beyond the grave. His will left a detailed line of succession, primarily excluding the Scottish Stuart family descended from his older (and somewhat scandal hit) sister Margaret.

The key to this succession plan was that Edward would rule with a protectorate council of 16, made up of equal numbers of Catholic conservatives and reformers. Should Edward die without issue (which of course, he did), then both Mary and Elizabeth were put back into the succession by Henry's will, followed by the Grey family (from his younger sister Mary).

Crucially though, they were never legitimised. They were in the succession, but never being legitimised they were bastards in legal terms. Henry clearly never saw a problem with this, probably presuming that Edward would live long enough to have his own heirs.

As Edward's illness became obviously terminal in 1553, he sought to copy his father by issuing his own plan of succession, which we call Edward's devise. This followed Henry's exactly, but with the exclusion of his two sisters due to their bastardy. This was primarily aimed at excluding Catholic Mary from the succession, who would be a mortal threat to the increasingly Protestant church his government under Northumberland and church under Cranmer had been building. However, eliminating Mary for her bastardy inevitably meant the same had to apply to Elizabeth; hence the shift to Frances Grey. However, Frances was already past childbearing age, having had three daughters and would therefore be unable to bare a male heir- seen as essential for political stability. To her fury, Edward, supported by Northumberland and Frances's husband, Suffolk, therefore changed his devise to skip her and go to her eldest daughter, Lady Jane and her male heirs- thus directly including Jane in the succession even prior to having any male children with her hastily married husband, Northumberland's son.

As we know, Mary "rebelled" and was able to seize control through a sizable army and the gradual drift of privy councillors to her cause as they saw the way the wind was blowing. With such support, her supposed bastardy simply was not an issue; the groundswell of popular support was based on the country effectively acknowledging that she was the rightful Queen, regardless of anything Henry VIII said about his marriage. Once she had restored papal control and full Catholicism by the end of 1554, there was little need for any legal legitimisation; marriage was a sacrament, indissoluble except by papal dispensation, which Henry had never had (in fact, Pope Clement had ruled the marriage valid in the mid 30's). Under restored Catholicism, Henry's marriage to Katherine had been dissolved only by her death and Mary was perfectly legitimate.

When Mary died and Elizabeth succeeded to the throne, there was equally no legislation to legitimise her. All religious legislation from 1529-53 had been repealed by Mary. Elizabeth did not restore this- her religion was created by her own acts of Uniformity, supremacy and injunctions. Her church of England was effectively a new entity- no decisions by Henry's church were necessarily valid. Therefore, there was considerable ambiguity as to whether her illegitimacy from Henry's reign remained legitimate.

Effectively, Elizabeth got round it over 45 years by simply never mentioning it or allowing anyone else to. Elizabeth very rarely, if ever, mentioned her mother- it was widely known as a banned topic. Deliberately so of course- Elizabeth did not want anyone to focus their minds on the ambiguity around her mother's marriage. Mary Stuart of course mentioned it plenty in her own persuit of support for her claim to the throne, but as a half French Catholic in a world where enemies were increasingly defined by religion, support for Elizabeth's right to the throne (and ignoring any questions around it) became a question of patriotism and upholding perceived religious (as in Protestant) freedom

Edit; typos and for clarity